I am reading a paper that wanted to analyze a real valued ordinary differential equations which is not analytically tractable, however, by transferring it to a complex-valued equation, the solution can be analytically obtained. Reference is here: https://arxiv.org/pdf/2105.04923.pdf
The real and complex-valued equations are below. My question is, whether the solution spaces of the following two systems (real version and complex version) are equivalent or one is covered by the other one?
$$\frac{d\theta_i}{dt}=\sum_{j=1,...,n}A_{ij}\sin(\theta_j-\theta_i)$$ $$\frac{d\theta_i}{dt}=\sum_{j=1,...,n}A_{ij}(\sin(\theta_j-\theta_i)-i\cos(\theta_j-\theta_i))$$
Note that the second equation is equivalent to $$\frac{d\vec{X}}{dt}=A\vec{X}$$ where $d\vec{X}=e^{i\vec\theta}$. This system has solution $\vec{X}=e^{tA}\vec{X}_0$.